Cartan Connections for Stochastic Developments on sub-Riemannian Manifolds

Complete Cartan Connections on Complex Manifolds

Complete complex parabolic geometries (including projective connections and conformal connections) are flat and homogeneous.

On Sublaplacians of Sub-riemannian Manifolds

In this note we address a notion of sublaplacians of sub-Riemannian manifolds. In particular for fat sub-Riemannian manifolds we answered the sublaplacian question proposed by R. Montgomery in [21, p.142]

Mass Transportation on Sub-Riemannian Manifolds

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann’s Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal m...

Connections in sub-Riemannian geometry

We introduce the notion of a connection over a bundle map and apply it to a sub-Riemannian geometry. It is shown that the concepts of normal and abnormal extremals of a sub-Riemannian structure can be characterized as parallel transported sections with respect to these generalized connections. Using this formalism we are able to give necessary and sufficient conditions for the existence of a sp...

Stochastic variational principle on riemannian manifolds